1. Sect. 10.3: Angular & Translational Quantities. Relations Between Them

2.  There MUST be relations
From circular motion:
A mass moving in a circle
has a translational (linear)
velocity v & a translational
(linear) acceleration a.
We’ve just seen that it also
has an angular velocity and
an angular acceleration.
 There MUST be relations
between the translational
& the angular quantities!

3. Connection Between Angular & Linear Quantities
Radians! 
v = (/t),  = rθ
 v = r(θ/t) = rω
v = rω
 Depends on r
(ω is the same for all points!)
v2 = r2ω2, v1 = r1ω1
v2 > v1 since r2 > r1

4. Relation Between Angular & Linear Velocity
v = (/t),  = rθ
 v = r (θ /t) = rω
v : depends on r
ω : the same for all points
v2 = r2ω2, v1 = r1ω1
v2 > v1

5. Relation Between Angular & Linear Acceleration
In direction of motion:
(tangential acceleration)
at = (dv/dt), v = rω
 at= r (dω/dt)
at = rα
at: depends on r
α : the same for all points
_________________

6. Angular & Linear Acceleration
From circular motion: there is also an acceleration  to
motion direction
(radial or centripetal
acceleration)
ac = (v2/r)
But v = rω
 ac= rω2
ac: depends on r
ω: the same for all points
_______________

7. Total Acceleration  Two  vector components of acceleration
Tangential:
at = rα
Radial:
ac= rω2
Total acceleration
= vector sum:
a = ac+ at
_________________
a ---

8. Total Acceleration NOTE!
The tangential component of the acceleration, at, is due to changing speed
The centripetal component of the acceleration, ac, is due to changing direction
The total acceleration can be found from these components with standard vector addition:

9. Relation Between Angular Velocity & Rotation Frequency
f = # revolutions / second (rev/s)
1 rev = 2π rad
 f = (ω/2π) or ω = 2π f = angular frequency
1 rev/s  1 Hz (Hertz)
Period: Time for one revolution.
 T = (1/f) = (2π/ω)

10. Translational-Rotational Analogues & Connections
Translation Rotation
Displacement x θ
Velocity v ω
Acceleration a α
CONNECTIONS
s = rθ, v = rω
at= r α
ac = (v2/r) = ω2 r

11. Example 10.2: CD Player = 1.8  105 radians = 2.8  104 revolutions
Consider a CD player playing a CD. For the player to
read a CD, the angular speed ω must vary to keep the
tangential speed constant (v = ωr). A CD has inner
radius ri = 23 mm = 2.3  10-2 m & outer radius
rf = 58 mm = 5.8  10-2 m. The tangential speed
at the outer radius is v = 1.3 m/s.
(A) Find angular speed in rev/min at inner radius:
ωi = (v/ri) = (1.3)/(2.3  10-2) = 57 rad/s = 5.4  102 rev/min
Outer radius: ωf = (v/rf) = (1.3)/(5.8  10-2) = 22 rad/s = 2.1  102 rev/min
(B) Standard playing time for a CD is 74 min, 33 s (= 4,473 s). How many revolutions does the disk make in that time?
θ = (½)(ωi + ωf)t = (½)( )(4,473 s)
= 1.8  105 radians = 2.8  104 revolutions